few_shots: Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001392

8
options: Use properties of operations to generate equivalent expressions.

Interpret the structure of expressions.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Use the four operations with whole numbers to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Work with radicals and integer exponents.

Understand the connections between proportional relationships, lines, and linear equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Create equations that describe numbers or relationships

Analyze patterns and relationships.

Write and interpret numerical expressions.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand properties of multiplication and the relationship between multiplication and division.

Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with equal groups of objects to gain foundations for multiplication.

Represent and solve problems involving addition and subtraction.

Use polynomial identities to solve problems.

Write expressions in equivalent forms to solve problems.

Perform arithmetic operations on polynomials.

Reason about and solve one-variable equations and inequalities.

Generate and analyze patterns.

Multiply and divide within 100.

Understand and apply properties of operations and the relationship between addition and subtraction.

Understand the relationship between zeros and factors of polynomials.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Solve systems of equations.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
problem_activity: Activity
In this activity, students first encounter exponential expressions with variables.
Launch
Recall with students that when we write
\(6x^2\)
, we mean to multiply 6 by the result of
\(x^2\)
. Remind them that the number part of such a product is called the
coefficient
of the expression, so in this example, 6 is the coefficient of
\(x^2\)
.
Give students 5 minutes of quiet work time to evaluate the expressions. Follow with whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have evaluated the first 2-3 expressions, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
Evaluate each expression for the given value of
\(x\)
.
\(3x^2\)
when
\(x\)
is 10
\(3x^2\)
when
\(x\)
is
\(\frac19\)
\(\frac{x^3}{4}\)
when
\(x\)
is 4
\(\frac{x^3}{4}\)
when
\(x\)
is
\(\frac12\)
\(9+x^7\)
when
\(x\)
is 1
\(9+x^7\)
when
\(x\)
is
\(\frac12\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may use the wrong order when evaluating expressions, such as in multiplying by 3 first in
\(3x^2\)
and then squaring. Have them recall the last lesson where they practiced applying the agreed on order of operations in numeric expressions.
Students may interpret an expression like
\(3x\)
as meaning 3 next to the digit
\(x\)
instead of as multiplication. For example, students may think that
\(3x\)
means 35 instead of 15 when
\(x=5\)
. Discuss how the shorthand notation of coefficients next to variables with no symbols between them tells us to multiply, and that it will simplify future work with expressions and equations.
Activity Synthesis
The purpose of the discussion is to ensure that students understand how to evaluate expressions with variables for a given value of the variable. It is also an opportunity for students to practice interpreting and using vocabulary like
coefficient, variable,
power,
and
exponent
.
Some guiding questions:
"In each expression, what is the coefficient?" (3, 3,
\(\frac14\)
,
\(\frac14\)
, 1, 1.)
"Choose one of the expressions. Describe the steps that you carried out to evaluate the expression." (Sample response: I rewrote
\(9+x^7\)
as
\(9+\left(\frac12\right)^7\)
. I needed to first evaluate the exponent. I knew that
\(\left(\frac12\right)^7\)
means
\(\frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12 \boldcdot \frac12\)
, which is
\(\frac{1}{128}\)
. So then I wrote
\(9 + \frac{1}{128}\)
, which can also be written
\(9\frac{1}{128}\)
.
"How is evaluating the expressions when
\(x\)
is a fraction similar to when
\(x\)
is a whole number?  How is it different?" (It's similar because you're still just multiplying
\(x\)
by itself a certain number of times. It's different because multiplying a fraction by a fraction is a bit more complicated than multiplying a whole number by itself.)
Speaking, Writing: MLR8 Discussion supports.
Use this to amplify mathematical uses of language to communicate about how to evaluate exponential expressions. Revoice the term “coefficient” during student discussion and press for detail when identifying the coefficients and the order in which they are multiplied in each expression.
Design Principle(s): Support sense-making


correct_option_index: 13